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When two particles interact, they can either form a bound state, or, they can scatter each other.
In the first case, the two particles always remain close to each other. In the latter case, however, the state of two particles ``looks like" that of two particles moving independently, like free particles, towards each other in ``remote past". They come closer, and interaction between them in no longer negligible. But after sufficiently long time the state of the combined system is a linear superposition of states each of which represents the pair of particles moving away from each other in various directions.
It is obvious that the interaction between the particles should be such that it goes to zero when they are well separated, and not a ``confining force" like that of harmonic force, for which only bound states exist. In general an interacting pair of particles will have both type of states.
In a large class of problems in non-relativistic theory, called ``potential scattering" there is a single particle in the presence of a potential which has finite value in some region called `scattering center', and which goes to zero at distances away from the center. In this case too, there can be bound states or the particle may behaves like a free particle in remote past moving towards the center, and again, at large times, is seen to be moving away from the center.
The subject matter of quantum theory of scattering is to find the probability of the specific final states given the initial state of the particles, and relate these probabilities to experimental quantities like cross-sections which are actually measured in laboratory. States of the combined system which do look like that of a pair of free particles ( or that of a single free particle in the case of potential scattering) in the remote past and future are called {\bf scattering states}. The harmless phrase ``states that look like that of a free particle in remote past" requires explanation. We give that in the next section.
